Chapter 6 — Applications of Derivatives — covers rate of change, tangents/normals, increasing/decreasing functions, maxima/minima, and approximations. Carries 8-10 marks.
Key Concepts
Rate of Change
dy/dx represents rate of change of y with respect to x.
If s = f(t) is displacement, then ds/dt = velocity, d²s/dt² = acceleration
If s = f(t) is displacement, then ds/dt = velocity, d²s/dt² = acceleration
Tangent and Normal
Slope of tangent at (x₁,y₁): m = dy/dx at (x₁,y₁)
Equation of tangent: y − y₁ = m(x − x₁)
Slope of normal = −1/m
Equation of normal: y − y₁ = (−1/m)(x − x₁)
Equation of tangent: y − y₁ = m(x − x₁)
Slope of normal = −1/m
Equation of normal: y − y₁ = (−1/m)(x − x₁)
Increasing and Decreasing Functions
f'(x) > 0 on (a,b) → f is strictly increasing on (a,b)
f'(x) < 0 on (a,b) → f is strictly decreasing on (a,b)
f'(x) = 0 at a point → possible turning point (check sign change)
f'(x) < 0 on (a,b) → f is strictly decreasing on (a,b)
f'(x) = 0 at a point → possible turning point (check sign change)
Maxima and Minima
First Derivative Test:
If f'(x) changes from + to − at x = c → local maximum
If f'(x) changes from − to + at x = c → local minimum
If no sign change → neither (inflection point)
Second Derivative Test:
At critical point (f'(c) = 0):
f”(c) < 0 → local maximum
f”(c) > 0 → local minimum
f”(c) = 0 → test fails (use first derivative test)
If f'(x) changes from + to − at x = c → local maximum
If f'(x) changes from − to + at x = c → local minimum
If no sign change → neither (inflection point)
Second Derivative Test:
At critical point (f'(c) = 0):
f”(c) < 0 → local maximum
f”(c) > 0 → local minimum
f”(c) = 0 → test fails (use first derivative test)
For absolute max/min on [a,b]: Find critical points in (a,b), evaluate f at critical points AND at endpoints a, b. Compare all values.
Approximations
f(x + Δx) ≈ f(x) + f'(x)·Δx
Quick Revision Points
- Tangent slope = dy/dx; Normal slope = −dx/dy
- f'(x) > 0 → increasing; f'(x) < 0 → decreasing
- Critical points: where f'(x) = 0 or doesn’t exist
- 1st derivative test: check sign change of f'(x)
- 2nd derivative test: f”(c) < 0 → max; f''(c) > 0 → min
- Absolute extrema on closed interval: check critical points + endpoints
Chapter Navigation
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Next: Integrals Class 12 Notes
Related Chapters in Class 12 Maths
- Continuity and Differentiability Class 12 Notes
- Integrals Class 12 Notes
- Differential Equations Class 12 Notes
Practice What You Learned
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